The deal was completed in the final hours of transfer deadline day, and came a day after Madrid signed Gareth Bale from Arsenal's arch rivals Tottenham Hotspur for a reported world-record fee in excess of £80 million.
Özil completed a medical in Munich on Monday afternoon and Arsenal subsequently announced on their website that he had signed a "long-term contract".
"We are extremely pleased to have signed Mesut Özil," said Arsenal manager Arsene Wenger. "He is a great player, with proven quality at both club and international level. We have watched and admired him for some time as he has all the attributes I look for in an Arsenal player."
Özil told the Arsenal website: "I am thrilled to be joining a club of the stature of Arsenal and am looking forward to playing in the Premier League. It will be great for my own personal development as a player and I am particularly looking forward to working with Arsene Wenger. From our negotiations it is clear the club has huge ambition and I look forward to being part of an exciting future."
The 24-year-old's arrival represents a huge coup for Wenger, who has come under pressure from the club's fans to sign new players during the current transfer window.
Traditionally reluctant to spend money, Wenger had only signed 20-year-old French forward Yana Sanogo from Auxerre and brought Mathieu Flamini back to the club following a spell at AC Milan before Özil's arrival.
Amid an ongoing eight-year trophy drought, that had fuelled accusations that the club lacked ambition, especially after the successive departures of Samir Nasri, Cesc Fabregas and Robin van Persie over the last two years.
But Özil's signing represents a serious statement of intent, particularly as Arsenal had never before spent £15 million on a single player.
"I have said throughout the summer that we have been working hard to bring in top quality players to this football club," Wenger added.
"This has involved many many people across the club and I have always had the full support of the club in making decisions on the football side."
Özil joined Madrid from Werder Bremen for a fee of around £12.4 million in August 2010 after shining in Germany's run to the semi-finals at that year's World Cup in South Africa.
The richly talented left-footer played a starring role as Madrid romped to the Liga title under Jose Mourinho in 2012, but his position at the Bernabeu was put under threat by the close-season signings of Isco and Bale.
A third-generation Turkish-German from Gelsenkirchen in west Germany, Özil began his career with local club Schalke before joining Bremen in 2008.
He has scored 14 goals for Germany in 47 appearances and becomes the most expensive German footballer in history.
He will encounter several familiar faces at the Emirates Stadium, with Germany colleagues Per Mertesacker and Lukas Podolski already on Arsenal's books.
Arsenal also announced the signing of Italian goalkeeper Emiliano Viviano, 27, from Palermo on a season-long loan.